控制与导航中的矩阵逼近

THE MATRIX APPROXIMATIONS IN CONTROL AND NAVIGATION

在系统与控制理论中,常遇到对称、半正定或正交矩阵,但由于种种原因会使它们失去这些特性。对任意给定的实矩阵A,在误差矩阵的欧几里德范数极小意义下,本文证明了矩阵A的最佳半正定及对称逼近的存在与唯一性,最佳正交逼近的存在性。并且A的最佳对称与最佳正交逼近分别为(A+A^T)/2及UV^T,其中U与V是A的奇异值分解式UDV^T中的正交阵。

Positive semi-definite, symmetric and orthogonal matrices are frequently encountered in practice. However, due to various causes, such as round off error in computation, these matrices may lose their characteristics. For an arbitrary matrix A ∈ Rn×n there exist its optimal orthogonal, symmetric and positive semi-definite matrix approximations, which minimize the norms of their error matrices. The above results are proved in this paper by means of the projection theorems in Hilbert space and the singular value decomposition of matrix. If a matrix A is written in its singular value decomposition form UDVT, its optimal symmetric: and orthogonal approximations are (A + AT)/2 and UVT, respectively.